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("tabex2" "\\begin{center} \\fbox{\\parbox{4.5in}{ \\begin{tabular}{ll} \\multicolumn{2}{c}{\\bf Box 2.1.1 \\quad Summary of Important Formulas for \\S 2.1} \\\\ {\\it Velocity\\/} & \\\\ & \\\\ $ V = {\\displaystyle \\frac{\\partial \\phi}{\\partial t} } $ & $ V ^a ={\\displaystyle \\frac{\\partial \\phi ^a} {\\partial t}}$ \\\\ & \\\\ $ v _t = V _t \\circ \\phi _t ^{-1} $ & $ v ^a _t = V ^a _t \\circ \\phi _t ^{-1} $ \\\\ & \\\\ {\\it Covariant Derivative\\/} & \\\\ & \\\\ $ {\\bf D} v \\cdot w = \\nabla _w v $ & $ (\\nabla _w v) ^a = {\\displaystyle \\frac{\\partial v ^a} {\\partial x ^b}} w ^b + \\gamma ^a _{b c} w ^b v ^c $ \\\\ \\end{tabular} }} \\end{center}" nil 0) ("tabex3" "\\begin{center} \\begin{tabular}{lll} {\\it Classical Tensor Analysis\\/} & & {\\it Tensor Analysis on Manifolds\\/} \\\\ \\\\ $ \\{ x ^a\\}$ & Coordinates & $ \\{ x ^a\\}$ \\\\ \\\\ $ e _a = {\\displaystyle \\frac{\\partial z ^i } {\\partial x ^a} \\dot{ i} _i }$ & $ \\!\\!\\! \\begin{array}{l} {\\rm coordinate} \\\\ {\\rm basis \\, vectors } \\end{array} $ & $ {\\displaystyle \\frac{\\partial}{\\partial x^a} = e _a } $ \\\\ \\\\ $ \\!\\!\\!\\! \\left. \\begin{array}{l} \\bar{e} _a = {\\displaystyle \\frac{\\partial x ^b} {\\partial \\bar{x} ^a} e _b } \\\\ \\\\ \\bar{e} ^a = {\\displaystyle \\frac{\\partial \\bar{x} ^a} {\\partial \\bar{x} ^b} e^b } \\end{array} \\right\\}$ & $ \\!\\!\\! \\begin{array}{l} {\\rm change \\, of} \\\\ {\\rm coordinates} \\\\ \\end{array} $ & $ \\left\\{ \\begin{array}{l} {\\displaystyle \\frac{\\partial}{\\partial \\bar{x} ^a} = \\frac{ \\partial x ^b}{\\partial \\bar{x} ^a} \\frac{ \\partial} {\\partial x ^b} } \\\\ \\\\ {\\displaystyle d \\bar{x} ^a = \\frac{ \\partial \\bar{x} ^a }{ \\partial x ^b} d x ^b } \\end{array} \\right. $ \\\\ \\end{tabular} \\end{center}" nil 0) ("tabex4" "\\begin{center} \\begin{tabular}{||l|l||} \\hline \\hline \\quad {\\bf Classical Mechanics} & \\quad {\\bf Quantum Mechanics} \\\\ \\hline \\hline immersed Lagrangian manifold & element of $L^2 (Q)$ or ${\\cal D}^\\prime (Q)$ \\\\ $\\Lambda \\rightarrow (T ^\\ast Q,\\Omega)$ & \\\\ \\hline $\\Lambda = $ graph of ${\\bf d} S$ & $\\psi = \\exp (iS/\\hbar)$ \\\\ \\hline $T ^{\\ast} Q$ & Hilbertspace \\\\ \\hline Lagrangian manifold & (possibly unbounded) \\\\ $\\quad \\Omega \\subset(T^\\ast Q, \\Omega _Q) \\times (T^\\ast R,-\\Omega_R)$ & $\\quad L^2 (R)$ to $L^2 (Q)$ \\\\ \\hline composition of canonical relations & composition of operators \\\\ \\hline \\hline \\end{tabular} \\end{center}" nil 0) ("tabex5" "\\begin{center} \\fbox{\\fbox{\\parbox{4.0in}{ \\begin{tabular}{ll} \\quad {\\bf Classical Mechanics} & \\quad {\\bf Quantum Mechanics} \\\\ \\hline \\hline immersed Lagrangian manifold & element of $L^2 (Q)$ or ${\\cal D}^\\prime (Q)$ \\\\ $\\Lambda \\rightarrow (T ^\\ast Q,\\Omega)$ & \\\\ \\hline $\\Lambda = $ graph of ${\\bf d} S$ & $\\psi = \\exp (iS/\\hbar)$ \\\\ \\hline $T ^{\\ast} Q$ & Hilbertspace \\\\ \\hline Lagrangian manifold & (possibly unbounded) \\\\ $\\quad \\Omega \\subset (T^\\ast Q, \\Omega _Q) \\times (T^\\ast R,-\\Omega_R)$ & $\\quad L^2 (R)$ to $L^2 (Q)$ \\\\ \\hline composition of canonical relations & composition of operators \\\\ \\end{tabular} }}} \\end{center}" nil 0) ("tabex6" "\\begin{center} \\begin{tabular}{||l|c|c||} \\hline Case & Conditions & Connection \\\\ \\hline Unconstrained & ${\\cal D}_q = T_q Q$ & ${\\cal A}^{\\rm sym}(\\dot q)={\\Bbb I} ^{-1} J(\\dot{q})$ \\\\ [1ex] Purely Kinematic & ${\\cal D}_q \\cap T_q({\\rm Orb}(q)) = \\{0\\}$ & ${\\cal A}^{\\rm kin} (\\dot{q}) = 0$ \\\\ [1ex] Horizontal symmetries & ${\\cal D}_q \\cap T_q({\\rm Orb}(q))_G = T_q({\\rm Orb}(q))_H$ & ${\\cal A}^{\\rm sym} (\\dot{q}) + {\\cal A}^{\\rm kin} (\\dot{q}) ={\\Bbb I} ^{-1} J_H (\\dot{q})$ \\\\ [1ex] General principal & ${\\cal D}_q + T_q({\\rm Orb}(q)) = T_q Q$ & ${\\cal A}^{\\rm sym}(\\dot{q}) + {\\cal A}^{\\rm kin}(\\dot{q}) = {\\Bbb I} ^{-1} J ^{\\rm nhc}(\\dot{q})$ \\\\ bundle case & & \\\\ 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[1982]}] {arnold:sing} Arnol$'$d, V. L., Varchenko, A. N., and Gusein-Zade, S. M. [1982] \\emph{Singularities of differentiable maps.} I, ``Nauka'', Moscow (Russian); English transl. Birkh\\\"auser, 1985. \\bibitem[{Arnol$'$d et al. 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